OG - DS - #109

[Guest]

When a rectangle is cut by a diagonal, are the two triangles formed "special" right triangles (i.e. 45/45/90) or just a regular right triangle?

Also the GMAC Quantitative Review states (Question #109 DS) that: "When the length of a diagonal of a rectangle is known, the Pythagorean theorem can "sometimes" be applied to determine the length or the width of the rectangle". However both the length and width cannot be dtermined if only the length of the diagonal is known.

Can someone please explain the "sometimes" application of this rule? In which situations can you determine the length or width?

Thanks!

[StaceyKoprince]

If you take the diagonal of a square, you have two 45-45-90 triangles. If you take the diagonal of a rectangle, you MIGHT have a 30-60-90 triangle... or you might have any other kind of right triangle.

Sketch a rectangle and draw in the diagonal. You know the triangles are right triangles, so you can use the Pythagorean theorem as long as you know 2 of the 3 sides. Or, if you know it's a 30-60-90, you can use the ratios of the sides, as long as you know the length of one side. Or, if you know the angles and the length of one side, you could use trig to get the other sides... but that last one you won't have to do on the GMAT (no trig on the GMAT0.

[Guest]

Thanks Stacey. That is very helpful.

I'm assuming the only way to know (on a typical GMAT problem) that the triangle is a 30-60-90 (when rectangle is cut by a diagonal), is if the GMAT tells you that one of the angles is 30, or 60 degrees?

[StaceyKoprince]

Yep, they'd either have to give you some info that would let you know one of those angles was 30 or 60, or they'd have to tell you the sides themselves so that you could see they fit the proper ratios... in which case you wouldn't care about using the angles to find the sides b/c you'd already have them. :)